NAG Toolbox: nag_ode_sl2_breaks_vals (d02kd)

Purpose

nag_ode_sl2_breaks_vals (d02kd) finds a specified eigenvalue of a regular or singular second-order Sturm–Liouville system on a finite or infinite interval, using a Pruefer transformation and a shooting method. Provision is made for discontinuities in the coefficient functions or their derivatives.

Description

nag_ode_sl2_breaks_vals (d02kd) finds a specified eigenvalue λ̃λ~ of a Sturm–Liouville system defined by a self-adjoint differential equation of the second-order

(p(x)y′)′ + q(x ; λ)y = 0, a < x < b,

(p(x)y′)′+q(x;λ)y=0, a<x<b,

together with appropriate boundary conditions at the two, finite or infinite, end points aa and bb. The functions pp and qq, which are real-valued, are defined by coeffn. The boundary conditions must be defined by bdyval, and, in the case of a singularity at aa or bb, take the form of an asymptotic formula for the solution near the relevant end point.

For the theoretical basis of the numerical method to be valid, the following conditions should hold on the coefficient functions:

(a)

p(x)p(x) must be nonzero and must not change sign throughout the interval (a,b)(a,b); and,

(b)

( ∂ q)/( ∂ λ)∂q∂λ must not change sign throughout the interval (a,b)(a,b) for all relevant values of λλ, and must not be identically zero as xx varies, for any λλ.

Points of discontinuity in the functions pp and qq or their derivatives are allowed, and should be included as ‘break points’ in the array xpoint.

The eigenvalue λ̃λ~ is determined by a shooting method based on the Scaled Pruefer form of the differential equation as described in Pryce (1981), with certain modifications. The Pruefer equations are integrated by a special internal function using Merson's Runge–Kutta formula with automatic control of local error. Providing certain assumptions (see Section [Timing]) are met, the computed value of λ̃λ~ will be correct to within a mixed absolute/relative error specified by tol.

A good account of the theory of Sturm–Liouville systems, with some description of Pruefer transformations, is given in Chapter X of Birkhoff and Rota (1962). An introduction to the use of Pruefer transformations for the numerical solution of eigenvalue problems arising from physics and chemistry is given in Bailey (1966).

Parameters

Compulsory Input Parameters

The points where the boundary conditions computed by bdyval are to be imposed, and also any break points, i.e., xpoint(1)xpoint1 to xpoint(m)xpointm must contain values x1, … ,xmx1,…,xm such that

x1 ≤ x2 < x3 < ⋯ < xm − 1 ≤ xm

x1≤x2<x3<⋯<xm-1≤xm

with the following meanings:

(a)

x1x1 and xmxm are the left- and right-hand end points, aa and bb, of the domain of definition of the Sturm–Liouville system if these are finite. If either aa or bb is infinite, the corresponding value x1x1 or xmxm may be a more-or-less arbitrarily ‘large’ number of appropriate sign.

(b)

x2x2 and xm − 1xm-1 are the Boundary Matching Points (BMPs), that is the points at which the left and right boundary conditions computed in bdyval are imposed.

If the left-hand end point is a regular point then you should set x2 = x1x2=x1( = a)(=a), while if it is a singular point you must set x2 > x1x2>x1. Similarly xm − 1 = xmxm-1=xm ( = b=b) if the right-hand end point is regular, and xm − 1 < xmxm-1<xm if it is singular.

Numerical integration of the differential equation is stopped and restarted at each break point. In simple cases no break points are needed. However, if p(x)p(x) or q(x ; λ)q(x;λ) are given by different formulae in different parts of the interval, then integration is more efficient if the range is broken up by break points in the appropriate way. Similarly points where any jumps occur in p(x)p(x) or q(x ; λ)q(x;λ), or in their derivatives up to the fifth-order, should appear as break points.

The index jj of the sub-interval ijij (see specification of xpoint) in which xx lies.

Output Parameters

1:
p – double scalar

The value of p(x)p(x) for the current value of xx.

2:
q – double scalar

The value of q(x ; λ)q(x;λ) for the current value of xx and the current trial value of λλ.

3:
dqdl – double scalar

The value of ( ∂ q)/( ∂ λ)(x ; λ)∂q∂λ(x;λ) for the current value of xx and the current trial value of λλ. However dqdl is only used in error estimation and, in the rare cases where it may be difficult to evaluate, an approximation (say to within 20%20%) will suffice.

3:
bdyval – function handle or string containing name of m-file

bdyval must define the boundary conditions. For each end point, bdyval must return (in yl or yr) values of y(x)y(x) and p(x)y′(x)p(x)y′(x) which are consistent with the boundary conditions at the end points; only the ratio of the values matters. Here xx is a given point (xl or xr) equal to, or close to, the end point.

For a regular end point (aa, say), x = ax=a, a boundary condition of the form

If aa is a regular end point of the system (so that a = x1 = x2a=x1=x2), then xl contains aa. If aa is a singular point (so that a ≤ x1 < x2a≤x1<x2), then xl contains a point xx such that x1 < x ≤ x2x1<x≤x2.

2:
xr – double scalar

If bb is a regular end point of the system (so that xm − 1 = xm = bxm-1=xm=b), then xr contains bb. If bb is a singular point (so that xm − 1 < xm ≤ bxm-1<xm≤b), then xr contains a point xx such that xm − 1 ≤ x < xmxm-1≤x<xm.

3:
elam – double scalar

The current trial value of λλ.

Output Parameters

1:
yl(33) – double array

yl(1)yl1 and yl(2)yl2 should contain values of y(x)y(x) and p(x)y′(x)p(x)y′(x) respectively (not both zero) which are consistent with the boundary condition at the left-hand end point, given by x = xlx=xl. yl(3)yl3 should not be set.

2:
yr(33) – double array

yr(1)yr1 and yr(2)yr2 should contain values of y(x)y(x) and p(x)y′(x)p(x)y′(x) respectively (not both zero) which are consistent with the boundary condition at the right-hand end point, given by x = xrx=xr. yr(3)yr3 should not be set.

4:
k – int64int32nag_int scalar

kk, the index of the required eigenvalue when the eigenvalues are ordered

An indication of the scale of the problem in the λλ-direction. delam holds the initial ‘search step’ (positive or negative). Its value is not critical, but the first two trial evaluations are made at elam and elam + delamelam+delam, so the function will work most efficiently if the eigenvalue lies between these values. A reasonable choice (if a closer bound is not known) is half the distance between adjacent eigenvalues in the neighbourhood of the one sought. In practice, there will often be a problem, similar to the one in hand but with known eigenvalues, which will help one to choose initial values for elam and delam.

If delam = 0.0delam=0.0 on entry, it is given the default value of 0.25 × max(1.0,|elam|)0.25×max(1.0,|elam|).

hmax(1,j)hmax1j should contain a maximum step size to be used by the differential equation code in the jjth sub-interval ijij (as described in the specification of parameter xpoint), for j = 1,2, … ,m − 3j=1,2,…,m-3. If it is zero the function generates a maximum step size internally.

It is recommended that hmax(1,j)hmax1j be set to zero unless the coefficient functions pp and qq have features (such as a narrow peak) within the jjth sub-interval that could be ‘missed’ if a long step were taken. In such a case hmax(1,j)hmax1j should be set to about half the distance over which the feature should be observed. Too small a value will increase the computing time for the function. See Section [Further Comments] for further suggestions.

The rest of the array is used as workspace.

9:
monit – function handle or string containing name of m-file

monit is called by nag_ode_sl2_breaks_vals (d02kd) at the end of each root-finding iteration and allows you to monitor the course of the computation by printing out the parameters (see Section [Example] for an example).

If no monitoring is required, the dummy (sub)program nag_ode_sl2_reg_finite_dummy_monit (d02kay) may be used. (nag_ode_sl2_reg_finite_dummy_monit (d02kay) is included in the NAG Toolbox.)

monit(nit, iflag, elam, finfo)

Input Parameters

1:
nit – int64int32nag_int scalar

The current value of the parameter maxit of nag_ode_sl2_breaks_vals (d02kd), this is decreased by one at each iteration.

The function is converging to the eigenvalue λ̃λ~ (having already bracketed it).

3:
elam – double scalar

The current trial value of λλ.

4:
finfo(1515) – double array

Information about the behaviour of the shooting method, and diagnostic information in the case of errors. It should not normally be printed in full if no error has occurred (that is, if iflag > 0iflag>0), though the first few components may be of interest to you. In case of an error (iflag < 0iflag<0) all the components of finfo should be printed.

An estimate of the quantity ∂λ∂λ defined as follows. Consider the perturbation in the miss-distance f(λ)f(λ) that would result if the local error in the solution of the differential equation were always positive and equal to its maximum permitted value. Then ∂λ∂λ is the perturbation in λλ that would have the same effect on f(λ)f(λ). Thus, at the zero of f(λ),|∂λ|f(λ),|∂λ| is an approximate bound on the perturbation of the zero (that is the eigenvalue) caused by errors in numerical solution. If ∂λ∂λ is very large then it is possible that there has been a programming error in coeffn such that qq is independent of λλ. If this is the case, an error exit with ifail = 5ifail=5 should follow. finfo(2)finfo2 is set to zero if iflag < 0iflag<0.

The number of internal iterations, using the same value of λλ and tighter accuracy tolerances, needed to bring the accuracy (that is, the value of ∂λ∂λ) to an acceptable value. Its value should normally be 1.01.0, and should almost never exceed 2.02.0.

The index of the sub-interval where failure occurred, in the range 11 to m − 3m-3. In case of an error in bdyval, it is set to 00 or m − 2m-2 depending on whether the left or right boundary condition caused the error.

The current values of the Pruefer dependent variables ββ, φϕ and ρρ respectively. These are set to zero in case of an error in bdyval. (See nag_ode_sl2_breaks_funs (d02ke) for a description of these variables.)

A bound on nrnr, the number of root-finding iterations allowed, that is the number of trial values of λλ that are used. If maxit ≤ 0maxit≤0, no such bound is assumed. (See also maxfun.)

Default:
00

3:
maxfun – int64int32nag_int scalar

A bound on nfnf, the number of calls to coeffn made in any one root-finding iteration. If maxfun ≤ 0maxfun≤0, no such bound is assumed.

Default:
00

Input Parameters Omitted from the MATLAB Interface

None.

Output Parameters

1:
elam – double scalar

The final computed estimate, whether or not an error occurred.

2:
delam – double scalar

If ifail = 0ifail=0, delam holds an estimate of the absolute error in the computed eigenvalue, that is |λ̃ − elam| ≃ delam|λ~-elam|≃delam. (In Section [General Description of the Algorithm] we discuss the assumptions under which this is true.) The true error is rarely more than twice, or less than a tenth, of the estimated error.

At some call to bdyval, invalid values were returned, that is, either yl(1) = yl(2) = 0.0yl1=yl2=0.0, or yr(1) = yr(2) = 0.0yr1=yr2=0.0 (a programming error in bdyval). See the last call of monit for details.

This error exit will also occur if p(x)p(x) is zero at the point where the boundary condition is imposed. Probably bdyval was called with xl equal to a singular end point aa or with xr equal to a singular end point bb.

The ‘bracketing’ phase (with parameter iflag of the monit equal to 11) failed to bracket the eigenvalue within ten iterations. This is caused by an error in formulating the problem (for example, qq is independent of λλ), or by very poor initial estimates of elam and delam.

On exit, elam and elam + delamelam+delam give the end points of the interval within which no eigenvalue was located by the function.

To obtain the desired accuracy the local error tolerance was set so small at the start of some sub-interval that the differential equation solver could not choose an initial step size large enough to make significant progress. See the last call of monit for diagnostics.

At some point inside a sub-interval the step size in the differential equation solver was reduced to a value too small to make significant progress (for the same reasons as with ifail = 7ifail=7). This could be due to pathological behaviour of p(x)p(x) and q(x ; λ)q(x;λ) or to an unreasonable accuracy requirement or to the current value of λλ making the equations ‘stiff’. See the last call of monit for details.

nag_roots_contfn_brent_rcomm (c05az), called by nag_ode_sl2_breaks_vals (d02kd), has terminated with the error exit corresponding to a pole of the residual function f(λ)f(λ). This error exit should not occur, but if it does, try solving the problem again with a smaller value for tol.

A serious error has occurred in an internal call. Check all (sub)program calls and array dimensions. Seek expert help.

Note: error exits with ifail = 2ifail=2, 33, 66, 77, 88 or 1111 are caused by being unable to set up or solve the differential equation at some iteration and will be immediately preceded by a call of monit giving diagnostic information. For other errors, diagnostic information is contained in
hmax(2,j)hmax2j, for j = 1,2, … ,mj=1,2,…,m, where appropriate.

Accuracy

Further Comments

Timing

The time taken by nag_ode_sl2_breaks_vals (d02kd) depends on the complexity of the coefficient functions, whether they or their derivatives are rapidly changing, the tolerance demanded, and how many iterations are needed to obtain convergence. The amount of work per iteration is roughly doubled when tol is divided by 1616. To make the most economical use of the function, one should try to obtain good initial values for elam and delam, and, where appropriate, good asymptotic formulae. Also the boundary matching points should not be set unnecessarily close to singular points.

General Description of the Algorithm

A shooting method, for differential equation problems containing unknown parameters, relies on the construction of a ‘miss-distance function’, which for given trial values of the parameters measures how far the conditions of the problem are from being met. The problem is then reduced to one of finding the values of the parameters for which the miss-distance function is zero, that is to a root-finding process. Shooting methods differ mainly in how the miss-distance is defined.

nag_ode_sl2_breaks_vals (d02kd) defines a miss-distance f(λ)f(λ) based on the rotation about the origin of the point p(x) = (p(x)y′(x),y(x))px=(p(x)y′(x),y(x)) in the Phase Plane as the solution proceeds from aa to bb. The boundary conditions define the ray (i.e., two-sided line through the origin) on which p(x)p(x) should start, and the ray on which it should finish. The eigenvaluekk defines the total number of half-turns it should make. Numerical solution is actually done by ‘shooting forward’ from x = ax=a and ‘shooting backward’ from x = bx=b to a matching point x = cx=c. Then f(λ)f(λ) is taken as the angle between the rays to the two resulting points Pa(c)Pa(c) and Pb(c)Pb(c). A relative scaling of the py′py′ and yy axes, based on the behaviour of the coefficient functions pp and qq, is used to improve the numerical behaviour.

Figure 1

The resulting function f(λ)f(λ) is monotonic over − ∞ < λ < ∞-∞<λ<∞, increasing if ( ∂ q)/( ∂ λ) > 0∂q∂λ>0 and decreasing if ( ∂ q)/( ∂ λ) < 0∂q∂λ<0, with a unique zero at the desired eigenvalue λ̃λ~. The function measures f(λ)f(λ) in units of a half-turn. This means that as λλ increases, f(λ)f(λ) varies by about 11 as each eigenvalue is passed. (This feature implies that the values of f(λ)f(λ) at successive iterations – especially in the early stages of the iterative process – can be used with suitable extrapolation or interpolation to help the choice of initial estimates for eigenvalues near to the one currently being found.)

The function actually computes a value for f(λ)f(λ) with errors, arising from the local errors of the differential equation code and from the asymptotic formulae provided by you if singular points are involved. However, the error estimate output in delam is usually fairly realistic, in that the actual error |λ̃ − elam||λ~-elam| is within an order of magnitude of delam.

The Position of the Shooting Matching Point c

This point is always one of the values xixi in array xpoint. It is chosen to be the value of that xixi, 2 ≤ i ≤ m − 12≤i≤m-1, that lies closest to the middle of the interval [x2,xm − 1][x2,xm-1]. If there is a tie, the rightmost candidate is chosen. In particular if there are no break points, then c = xm − 1c=xm-1 ( = x3=x3); that is, the shooting is from left to right in this case. A break point may be inserted purely to move cc to an interior point of the interval, even though the form of the equations does not require it. This often speeds up convergence especially with singular problems.

Here q(x ; λ)q(x;λ) is nearly constant over the range except for a sharp inverted spike over approximately − 0.1 ≤ x ≤ 0.1-0.1≤x≤0.1. There is a danger that the function will build up to a large step size and ‘step over’ the spike without noticing it. By using break points – say at ± 0.5±0.5 – one can restrict the step size near the spike without impairing the efficiency elsewhere.

Examples of Boundary Conditions at Singular Points

Quoting from page 243 of Bailey (1966): ‘Usually ... the differential equation has two essentially different types of solution near a singular point, and the boundary condition there merely serves to distinguish one kind from the other. This is the case in all the standard examples of mathematical physics.’

In most cases the behaviour of the ratio p(x)y′ / yp(x)y′/y near the point is quite different for the two types of solution. Essentially what you provide through the bdyval is an approximation to this ratio, valid as xx tends to the singular point (SP).

You must decide (a) how accurate to make this approximation or asymptotic formula, for example how many terms of a series to use, and (b) where to place the boundary matching point (BMP) at which the numerical solution of the differential equation takes over from the asymptotic formula. Taking the BMP closer to the SP will generally improve the accuracy of the asymptotic formula, but will make the computation more expensive as the Pruefer differential equations generally become progressively more ill-behaved as the SP is approached. You are strongly recommended to experiment with placing the BMPs. In many singular problems quite crude asymptotic formulae will do. To help you avoid needlessly accurate formulae, nag_ode_sl2_breaks_vals (d02kd) outputs two ‘sensitivity coefficients’
σl,σrσl,σr which estimate how much the errors at the BMPs affect the computed eigenvalue. They are described in detail in Section [The Sensitivity s and ].

the boundary conditions being that yy should remain bounded as xx tends to 00 and xx tends to ∞∞.

At the end x = 0x=0 there is one solution that behaves like x2x2 and another that behaves like x − 1x-1. For the first of these solutions p(x)y′ / yp(x)y′/y is asymptotically 2 / x2/x while for the second it is asymptotically − 1 / x-1/x. Thus the desired ratio is specified by setting

At the end x = ∞x=∞ the equation behaves like Airy's equation shifted through λλ, i.e., like y ′ ′ − ty = 0y′′-ty=0 where t = x − λt=x-λ, so again there are two types of solution. The solution we require behaves as

exp( − (2/3)t(3/2)) / root(4,t)

exp(-23t32)/t4

and the other as

exp( + (2/3)t(3/2)) / root(4,t).

exp(+23t32)/t4.

Hence, the desired solution has p(x)y′ / y ∼ − sqrt(t)p(x)y′/y∼-t so that we could set yr(1) = 1.0yr1=1.0 and yr(2) = − sqrt(x − λ)yr2=-x-λ. The complete function might thus be

Clearly for this problem it is essential that any value given by nag_ode_sl2_breaks_vals (d02kd) to xr is well to the right of the value of elam, so that you must vary the right-hand BMP with the eigenvalue index kk. One would expect λkλk to be near the kkth zero of the Airy function Ai(x)Ai(x), so there is no problem estimating elam.

More accurate asymptotic formulae are easily found: near x = 0x=0 by the standard Frobenius method, and near x = ∞x=∞ by using standard asymptotics for Ai(x)Ai(x), Ai′(x)Ai′(x), (see page 448 of Abramowitz and Stegun (1972)).

For example, by the Frobenius method the solution near x = 0x=0 has the expansion

The Sensitivity Parameters σl and σr

The sensitivity parameters σlσl, σrσr (held in hmax(1,m − 1)hmax1m-1 and hmax(1,m)hmax1m on output) estimate the effect of errors in the boundary conditions. For sufficiently small errors ΔyΔy, Δpy′Δpy′ in yy and py′py′ respectively, the relations

Δλ ≃ (y . Δpy′ − py′ . Δy)lσl

Δλ ≃ (y . Δpy′ − py′ . Δy)rσr

Δλ≃(y.Δpy′-py′.Δy)lσlΔλ≃(y.Δpy′-py′.Δy)rσr

are satisfied, where the subscripts ll, rr denote errors committed at the left- and right-hand BMPs respectively, and ΔλΔλ denotes the consequent error in the computed eigenvalue.

‘Missed Zeros’

This is a pitfall to beware of at a singular point. If the BMP is chosen so far from the SP that a zero of the desired eigenfunction lies in between them, then the function will fail to ‘notice’ this zero. Since the index of kk of an eigenvalue is the number of zeros of its eigenfunction, the result will be that

(a)

the wrong eigenvalue will be computed for the given index kk – in fact some λk + k′λk+k′ will be found where k′ ≥ 1k′≥1;

(b)

the same index kk can cause convergence to any of several eigenvalues depending on the initial values of elam and delam.

It is up to you to take suitable precautions – for instance by varying the position of the BMPs in the light of knowledge of the asymptotic behaviour of the eigenfunction at different eigenvalues.